On $ \sigma $-subnormal subgroups and products of finite groups

A. A. Heliel; A. Ballester-Bolinches; M. M. Al-Shomrani; R. A. Al-Obidy; A. A. Heliel; A. Ballester-Bolinches; M. M. Al-Shomrani; R. A. Al-Obidy

doi:10.3934/era.2023038

Electronic Research Archive

2023, Volume 31, Issue 2: 770-775. doi: 10.3934/era.2023038

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Research article

On $ \sigma $-subnormal subgroups and products of finite groups

1.
Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah, Saudi Arabia
2.
Departament de Matemàtiques, Universitat de València, Valencia, Spain

Received: 13 October 2022 Revised: 09 November 2022 Accepted: 13 November 2022 Published: 23 November 2022

Suppose that $ \sigma = \{ {\sigma}_{i} : i \in I \} $ is a partition of the set $ \mathbb{P} $ of all primes. A subgroup $ A $ of a finite group $ G $ is said to be $ \sigma $-subnormal in $ G $ if $ A $ can be joined to $ G $ by a chain of subgroups $ A = A_{0} \subseteq A_{1} \subseteq \cdots \subseteq A_{n} = G $ such that either $ A_{i-1} $ normal in $ A_{i} $ or $ A_{i}/ Core_{A_{i}}(A_{i-1}) $ is a $ {\sigma}_{j} $-group for some $ j \in I $, for every $ 1\leq i \leq n $. A $ \sigma $-subnormality criterion related to products of subgroups of finite $ \sigma $-soluble groups is proved in the paper. As a consequence, a characterisation of the $ \sigma $-Fitting subgroup of a finite $ \sigma $-soluble group naturally emerges.
- finite group,
- $ \sigma $-soluble group,
- $ \sigma $-subnormal subgroup,
- products of groups
Citation: A. A. Heliel, A. Ballester-Bolinches, M. M. Al-Shomrani, R. A. Al-Obidy. On $ \sigma $-subnormal subgroups and products of finite groups[J]. Electronic Research Archive, 2023, 31(2): 770-775. doi: 10.3934/era.2023038

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Abstract

Suppose that $ \sigma = \{ {\sigma}_{i} : i \in I \} $ is a partition of the set $ \mathbb{P} $ of all primes. A subgroup $ A $ of a finite group $ G $ is said to be $ \sigma $-subnormal in $ G $ if $ A $ can be joined to $ G $ by a chain of subgroups $ A = A_{0} \subseteq A_{1} \subseteq \cdots \subseteq A_{n} = G $ such that either $ A_{i-1} $ normal in $ A_{i} $ or $ A_{i}/ Core_{A_{i}}(A_{i-1}) $ is a $ {\sigma}_{j} $-group for some $ j \in I $, for every $ 1\leq i \leq n $. A $ \sigma $-subnormality criterion related to products of subgroups of finite $ \sigma $-soluble groups is proved in the paper. As a consequence, a characterisation of the $ \sigma $-Fitting subgroup of a finite $ \sigma $-soluble group naturally emerges.

References

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